Applications Of The Martingale And Empirical Process Methods In Statistics

Using the martingale and empirical process methods, this dissertation in-vestigates the nonparametric statistics and limit theorems for dependent obser-vations, many results are constructed. The main contents include the following aspects.Firstly, Using the martingale method and blocking technique, the central limit theorems (CLT) for the kernel density estimator fn,K(x) and the distribu-tion function estimator Fn,K(x)=f-∞x fn,K(t)dt are established. Further,we construct a moment inequality of the conditional expectation for dependent ran-dom variables. As applications, the convergence rates of‖fn,K(x)-Efnt,Kx(x)‖p in sup-norm loss and integral Lp-norm loss are proved. Moreover, the a.s. con-vergence rates of the supremum of|fn,K(x)-E fn,K(x)|over a compact set and the whole real line are obtained. It is showed, under suitable conditions, that the optimal rates for i.i.d. random variables are also optimal for dependent ones.Secondly, using the blocking technique, we construct m dependent random variables, then we reduce the investigation for dependent samples to that for independent ones. Several limit theorems are established:First, the pointwise and the uniformly optimal weak convergence rates of the deviation of the kernel density estimator with respect to its mean (and the true density function) are derived. Moreover, the pointwise and the uniformly optimal strong convergence rates are obtained.Thirdly, we establish the CLT for wavelet regression estimator. Moreover, using empirical process methods, the upper bound for the strong expectation with respect to the entropy, Bernstein inequality and Talagrand inequality, we derive the a.s. best possible convergence rate in sup-norm loss for wavelet regression estimators.Fourthly, using the martingale method and blocking technique, we establish strong invariance principles for sums of stationary ρ-mixing random variables with finite and unbounded second moments under weaker mixing rates. Some earlier results are improved. As applications, some LIL results with finite and unbounded variance are obtained, also a conjecture raised by Shao (1993a) is solved.Finally, using the probability inequalities and the weak invariance principles, the limit behavior of the complete moment convergence of partial sums and maximal partial sums for negatively associated random variables is investigated. Some more general and new results are given.